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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: MONDAY 10, JUNE 2013

COURSE CODE: M337/B

Question 1

Determine each of the following complex numbers in Cartesian form, simplifying your answers as far as possible.

(a)  

(b) the principal cube root of -8

(c)

(d)

ANSWERS(Purchase full paper to get all the solution):

1a)

 

 

               =  

               =

               =

Therefore,

 

1b)

the principal cube root of -8

 

Using De moivre’s theorem

 

On the horizontal

 

 

 

by comparing

 

 

This implies that r =2

 

the principal cube root of -8

 

          = 

the principal cube root of -8 = 

1c)

 

 

 

 

                =

 

1d)

 

 

                       =

 

Question 2

Let

A = {z : 1<z≤4 } and B = {z : Rez>-1,>Im z > -1}.

(a) Make separate sketches of the sets

A, B, C = A - B and D = ∂B.

(b) For each of the sets A, B, C and D:

(i) state whether it is a region, and if not explain why not

(ii) state whether it is compact, and if not explain why not.

Question 3

In this question Γ is the circle {z : |z| = 2}.

(a) Write down the standard parametrization for Γ.

(b) Evaluate

.

(c) Determine an upper estimate for the modulus of

 

Question 4

Evaluate the following integrals in which C = {z : |z| = 1}.

Name any standard results that you use and check that their hypotheses are satisfied.

1. z
2. 
3. 

Question 5

(a) Find the residues of the function

 

at each of the poles of f.

(b) Hence evaluate the integrals

                            

Question 6

Let f(z) =  + 5.

(a) Determine the number of zeros of f that lie inside:

(i) the circle C₁ = {z : |z| = 2},

(ii) the circle C₂ = {z : |z| = 1}.

(b) Show that the equation  + 5 = 0 has exactly 3 solutions in the set {z : 1 < |z| < 2}

Question 7

Let q(z) =  be a velocity function.

(a) Explain why q represents a model fluid flow on .

(b) Determine a complex potential function for this flow. Hence sketch the streamline through the point 1 and indicate the direction of flow.

(c) Evaluate the circulation of q along the path

Γ : γ(t) = t (t ∈ [0, 1]).

Question 8

(a) Show that the iteration sequence

       

  is conjugate to the iteration sequence

        =    , n = 0, 1, 2,...,

with .

(b) Find the fixed points of  and determine their nature.

(c) Determine whether or not lies in the Mandelbrot set M.

Question 9

(a) Let f be the function

.

i) Write  where u and v are real-valued functions.

(ii) Use the Cauchy–Riemann theorem and its converse to show that f is differentiable, at but not analytic there.

 

(b) Let g be the function .

(i) Show that g is conformal at 1.

(ii) Describe the effect of g on a small disc centred at 1.

(iii) Let  and  be the paths

:  =  

 : ) =

Show that  and  intersect at the points 1, and and find the angle from  to  at this point of intersection

(iv) Sketch the paths  and  on the same diagram, clearly indicating their directions.

(v) Using part (b)(ii), or otherwise, sketch the directions of g() and g() at g(1).

 vi) Find the image of the unit circle under g. Why is g not conformal at ?

Question 10

(a) Let f be the function

 

(i) Locate and classify the singularities of f. 

(ii) Find the Laurent series about 2 for f on the annulus

{z : 1 < |z - 2| < 3},

giving the constant term and two terms on each side of it.

(b) (i) Find the Taylor series about 0 (up to the term in ) for the function

g(z) =  

and explain why the series represents g on C.

(ii) Hence evaluate the integral

,

where C is the circle {z : |z| = 2

Question 11

Let f be the function

 .

(a) (i) Find the residues of f at each of the points 0,  and .

(b) Hence determine the sum of the series

 

(c) Use your solution to part (a)(ii) to prove that

           

Question 12

(a) Determine the extended Möbius transformation  that maps

 -1 to 0, ∞ to 1  and -i to ∞

(b) Let

                            R = {z : |z | < 1}∩{z : |z +1+ i| < 1},

                            S = {z₁ : 3π/2< Arg_2π(z₁) < 5π/4},

                            T = {w : Rew > 0}

(i) Sketch the regions R, S and T.

(ii) Explain why  (R) = S.

((iii) Hence determine a one-one conformal mapping f from R onto T.

(iv) Write down the rule of the inverse function .

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